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New maths proof shows how to stack oranges in 24 dimensions

By Lisa Grossman

IT’S a tight squeeze. Mathematicians have proved that they know the best way to pack spheres in eight and 24 dimensions – the first time this problem has been solved in a new dimension for almost 20 years.

“I think they’re fantastic results. I’m excited that this has been done at last,” says Thomas Hales at the University of Pittsburgh, Pennsylvania.

The sphere-packing problem asks a deceptively simple question: what arrangement crams the most spheres into a limited volume? It is easy to describe, but difficult to prove.

In 1611, Johannes Kepler suggested that the best arrangement for 3-dimensional spheres like oranges is a pyramid. But it took until 1998 for Hales to publish a proof – and another 16 years to formally verify it.

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Meanwhile, mathematicians have been gunning for higher dimensions. Now, Maryna Viazovska at Humboldt University of Berlin has proved that a grid called the E8 lattice is the best packing in eight dimensions (arxiv.org/abs/1603.04246v1). Almost immediately after, she teamed up with other researchers to prove that a related arrangement called the Leech lattice is best in 24 dimensions (arxiv.org/abs/1603.06518v1).

The fact that these dimensions were next to fall is no coincidence. For reasons we don’t understand, such lattices don’t show up in other dimensions. But they were suspected to be the most efficient arrangements in the dimensions they apply to.

“These are unbelievably good packings,” says Henry Cohn at Microsoft Research New England in Cambridge, Massachusetts. “The spheres in these dimensions fit perfectly, in ways that don’t happen in other dimensions.”

Packing spheres in 24 dimensions isn’t just a mathematical game. The problem has applications in wireless communication, and has been used to communicate with spacecraft in the distant solar system.

This article appeared in print under the headline “How to stack oranges in 24 dimensions”